OFFSET
2,1
COMMENTS
From A.H.M. Smeets, Jun 19 2019: (Start)
In the following, dig(expr) stands for the digit that represents the value of expression expr, and . stands for concatenation.
As for the naming of this sequence, the trivial 1 digit palindromes 0..dig(n-1) are excluded.
If a number m is palindromic in bases n and n+1, then m has an odd number of digits when represented in base n.
All three digit numbers in base n, that are palindromic in bases n and n+1 are given by:
101_3 22_4 for n = 3,
232_n 1.dig(n).1_(n+1)
343_n 2.dig(n-1).2_(n+1)
up to and including
dig(n-2).dig(n-1).dig(n-2)_n dig(n-3).4.dig(n-3)_(n+1) for n > 3, and
dig(n-1).0.dig(n-1)_n dig(n-3).5.dig(n-3)_(n+1) for n > 4.
Let d_L(n) be the number of integers with L digits in base n (L being odd), being palindromic in bases n and n+1, then:
d_1(n) = n for n >= 2 (see above),
d_3(n) = n-2 for n >= 5 (see above),
d_5(n) = n-1 for n >= 7 and n == 1 (mod 3),
d_5(n) = n-4 for n >= 7 and n in {0, 2} (mod 3), and
it seems that d_7(n) is of order O(n^2*log(n)) for n large enough. (End)
LINKS
Colin Barker, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
a(n) = 2n^2 + 3n + 2 for n >= 4 (which is 232_n and 1n1_(n+1)).
a(n) = A130883(n+1) for n > 3. - Robert G. Wilson v, Oct 08 2014
From Colin Barker, Jun 30 2019: (Start)
G.f.: x^2*(6643 - 19919*x + 19945*x^2 - 6684*x^3 + 19*x^4) / (1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>6.
(End)
EXAMPLE
a(14) = 2*14^2 + 3*14 + 2 = 436, which is 232_14 and 1e1_15.
MATHEMATICA
Do[ k = n + 2; While[ RealDigits[ k, n + 1 ][ [ 1 ] ] != Reverse[ RealDigits[ k, n + 1 ][ [ 1 ] ] ] || RealDigits[ k, n ][ [ 1 ] ] != Reverse[ RealDigits[ k, n ][ [ 1 ] ] ], k++ ]; Print[ k ], {n, 2, 75} ]
palQ[n_Integer, base_Integer] := Block[{idn = IntegerDigits[n, base]}, idn == Reverse[idn]]; f[n_] := Block[{k = n + 2}, While[ !palQ[k, n] || !palQ[k, n + 1], k++ ]; k]; Table[ f[n], {n, 2, 48}] (* Robert G. Wilson v, Sep 29 2004 *)
PROG
(PARI) isok(j, n) = my(da=digits(j, n), db=digits(j, n+1)); (Vecrev(da)==da) && (Vecrev(db)==db);
a(n) = {my(j = n); while(! isok(j, n), j++); j; } \\ Michel Marcus, Nov 16 2017
(PARI) Vec(x^2*(6643 - 19919*x + 19945*x^2 - 6684*x^3 + 19*x^4) / (1 - x)^3 + O(x^50)) \\ Colin Barker, Jun 30 2019
CROSSREFS
KEYWORD
nonn,easy,base
AUTHOR
Ulrich Schimke (ulrschimke(AT)aol.com)
EXTENSIONS
More terms from Robert G. Wilson v, Aug 14 2000
STATUS
approved